Björklund, Andreas
[VerfasserIn];
Kaski, Petteri
[VerfasserIn];
Koutis, Ioannis
[VerfasserIn]
;
Andreas Björklund and Petteri Kaski and Ioannis Koutis
[MitwirkendeR]
Directed Hamiltonicity and Out-Branchings via Generalized Laplacians
Anmerkungen:
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Beschreibung:
We are motivated by a tantalizing open question in exact algorithms: can we detect whether an n-vertex directed graph G has a Hamiltonian cycle in time significantly less than 2^n? We present new randomized algorithms that improve upon several previous works: 1. We show that for any constant 0<lambda<1 and prime p we can count the Hamiltonian cycles modulo p^((1-lambda)n/(3p)) in expected time less than c^n for a constant c<2 that depends only on p and lambda. Such an algorithm was previously known only for the case of counting modulo two [Bj\"orklund and Husfeldt, FOCS 2013]. 2. We show that we can detect a Hamiltonian cycle in O^*(3^(n-alpha(G))) time and polynomial space, where alpha(G) is the size of the maximum independent set in G. In particular, this yields an O^*(3^(n/2)) time algorithm for bipartite directed graphs, which is faster than the exponential-space algorithm in [Cygan et al., STOC 2013]. Our algorithms are based on the algebraic combinatorics of "incidence assignments" that we can capture through evaluation of determinants of Laplacian-like matrices, inspired by the Matrix--Tree Theorem for directed graphs. In addition to the novel algorithms for directed Hamiltonicity, we use the Matrix--Tree Theorem to derive simple algebraic algorithms for detecting out-branchings. Specifically, we give an O^*(2^k)-time randomized algorithm for detecting out-branchings with at least k internal vertices, improving upon the algorithms of [Zehavi, ESA 2015] and [Bj\"orklund et al., ICALP 2015]. We also present an algebraic algorithm for the directed k-Leaf problem, based on a non-standard monomial detection problem.